Exercise 45 Page 319

Recall that the recursive equation for an arithmetic sequence is of the following form.

a_{n} = a_{n - 1} + d

Here

d

is the common difference. Then, for the given sequence, the common difference is

- 12 .

a_{n} = a_{n - 1} - 12 \Leftrightarrow a_{n} = a_{n - 1} + (- 12)

The explicit formula for the sequence, with first term

a_{1} = 6

and common difference

- 12,

can be written as follows.

a_{n} = a_{1} + (n - 1) d \Rightarrow a_{n} = 6 + (n - 1) (- 12)

Let's simplify it.

a_{n} = 6 + (n - 1) (- 12)

Distribute $(- 12)$

a_{n} = 6 - 12 n + 12

Add terms

a_{n} = 18 - 12 n

The explicit rule for the sequence is

a_{n} = 18 - 12 n .

Correct : Incorrect : a_{n} = 18 - 12 n ✓ a_{n} = - 6 + 12 n \times